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Using Continuity of a Trig. Function to Rewrite It

  1. Nov 30, 2012 #1
    I used Wolfram Alpha to evaluate:

    [itex]lim tan[(2nπ)/(1 + 8n)][/itex]
    n->infinity

    it says that it can use the continuity of tan(n) at n = π / 4 to rewrite the aforementioned function as:

    [itex]tan[lim ((2nπ)/(1 + 8n))][/itex]
    n->infinity

    What is it talking about? I was taught to use certain properties of trig functions as they pertain to limits to solve limits of trig. functions, but this is a bit beyond me.

    P.S. I'm not using WA to do my homework or anything, I just wanted to see how one goes about solving a trig. limit like this, as I felt that it wasn't very straighforward.
     
  2. jcsd
  3. Nov 30, 2012 #2

    LCKurtz

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    Using the variable ##x## in stead of ##n##, look at the limit of$$
    \lim_{x\rightarrow \infty}\frac {2\pi x}{1+8x}$$What does that converge to? That will show you what they are talking about.
     
  4. Nov 30, 2012 #3
    I can see that it converges to π / 4, and I can prove it be substituting some things and simplifying it that way. But I don't understand why they can just at the very beginning of the problem rewrite it in that way from the get go.
     
  5. Nov 30, 2012 #4

    LCKurtz

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    When you are dealing with continuous functions, remember that, to put it loosely, "the limit of the function is the function of the limit". That is what allows you to take the limit "across" the function as in$$
    \lim_{x\rightarrow a}f(\hbox{anything}) = f(\lim_{x\rightarrow a}\hbox{anything})$$as long as the inside limit works. So if you can figure out the limit of the inside part, you are home free.
     
  6. Nov 30, 2012 #5
    Alright then. I gotcha. Thanks, I do appreciate it.
     
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